According the the nLab, the category of compactly generated (CG) spaces is not locally cartesian closed. So if $A$ is a CG space and $C$ a CG space above $A$, $C$ may not be exponentiable.
What if we require that $C$ is fibrant over $A$?
If $C\to A$ is a (Serre) fibration, is $C$ exponentiable in the category of CG spaces above $A$?
